M345t2CS11Sol - Embry-Riddle Aeronautical University MA 345...

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Embry-Riddle Aeronautical University C. Jacobs MA 345 Di ff erential Equations Exam II Solutions Spring 2011 Problems 1 - 5. (20 points) Match each di ff erential equation to its solution by putting the appropriate letter next to the equation number. 1) ( D - 4) 2 y = 0 a ) y = c 1 e 2 x + c 2 e - 2 x 2) ( D 2 - 4 ) y = 0 b ) y = c 1 e 2 x + c 2 xe 2 x 3) ( D 2 + 4 ) y = 0 c ) y = c 1 + c 2 e 4 x 4) ( D 2 - 4 D + 4 ) y = 0 d ) y = c 1 e 4 x + c 2 xe 4 x 5) ( D 2 - 4 D ) y = 0 e ) y = c 1 cos(2 x ) + c 2 sin(2 x ) Solutions: 1 - d) 2 - a) 3 - e) 4 - b) 5 - c) 6. (5 points) Let A be a 2 by 2 matrix and let X = x 1 x 2 . Suppose X = ( 0 0 ) is the only solution of AX = ( 0 0 ) . Then, which of the following statements must be true? a ) A - 1 exists b ) det( A ) = 0 c ) A must be ( 0 0 0 0 ) d ) A must be ( 1 0 0 1 ) e ) none of these 7. (5 points) In the solution of ( D 2 - D ) y = x , the general form of the particular solution y p is: a ) b 1 x b ) b 1 + b 2 x c ) b 1 x + b 2 x 2 d ) b 1 + b 2 e x e ) none of these 8. (5 points) Suppose z = i . Then z is equal to: a ) e i π 2 b ) e i π c ) e i 3 π 2 d ) e i 2 π e ) none of these 9. (5 points) ( D - 1) 2 ( e x sin x ) equals: a ) e x sin x b ) e x cos x c ) - e x sin x d ) 0 e
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